This gives the following equations for an ephemeris:
Heliocentric coordinates relating to the ecliptic.
The differences of the values of from their mean indicate the residual errors of this hypothesis. They indicate differences in the calculated and the observed geocentric positions which are represented by the geocentric angles subtended by the path described by the planet in the following fractions of a day: .00054, .00003, .00052. Since the heliocentric motion of the planet is about one-fourth of a degree per day, and the planet is considerably farther from the earth than from the sun at the times of the first and third observations, the errors will be less than half a second in arc.
If we desire all the accuracy possible with seven-figure logarithms, we may form a third hypothesis based on the following corrections:
The equations for an ephemeris will then be:
Heliocentric coordinates relating to the ecliptic.
The agreement of the calculated geocentric positions with the data is shown in the following table:
=E_{\text{in seconds}}-[4.2216270]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b380b7014a3dc20a4fd561050d4b54ce0ea95dc)
![{\displaystyle x=+.1820700-[0.3530366]\cos E-[0.1827457]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5129f4a74f3c363fa3539411fddd03252b112c66)
![{\displaystyle y=-.1244685+[0.1878576]\cos E-[0.3603257]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d09a01895195c1e1c0797f56fe6764c580dc8de)
![{\displaystyle z=-.03739370+[9.6656346]\cos E+[9.3320292]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1def17ff7375959ff32378768784b7e37d985b67)
=E_{\text{in seconds}}-[4.2216530]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76234b8bdbc2724191dd3495f4ccc6b1371f89b8)
![{\displaystyle x=+.1820765-[0.3530261]\cos E-[0.1827783]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1c6c992d3c832be38e323f53238f37bbe3c7b5)
![{\displaystyle y=-.1244853+[0.1878904]\cos E-[0.3603153]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf0886523112d8c42e4e5d89b36d891f3b101b4)
![{\displaystyle z=-.03739987+[9.6656285]\cos E+[9.3320758]\sin E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26183186824db35512c888311cf95b7a0db939c9)
